Cofibration

Summary

A cofibration is a continuous map $i \colon A \to X$ along which homotopies extend: it has the homotopy extension property for every target space. Inclusions of subcomplexes of CW complexes are the main examples. Cofibrations are well behaved for gluing, and for a cofibration the quotient $X / A$ has the homotopy type of the mapping cone. They are dual, in the sense of Eckmann–Hilton duality, to fibrations.